Multi-Objective Design and Selection of One Single Optimal Solution

2004 ◽  
Author(s):  
F. Levi ◽  
M. Gobbi ◽  
M. Farina ◽  
G. Mastinu
Keyword(s):  
Optimal Solution ◽  
Engineering Problem ◽  
Design Solution ◽  
Pareto Optimal ◽  
Large Set ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Pareto Optimal Set ◽  
Final Design ◽  
Selection Of

In the paper, the problem of choosing a single final design solution among a large set of Pareto-optimal solutions is addressed. Two methods, the k-optimality approach and the more general k-ε-optimality method will be introduced. These two methods theoretically justify and mathematically define the designer’s tendency to choose solutions which are “in the middle” of the Pareto-optimal set. These two methods have been applied to the solution of a relatively simple engineering problem, i.e. the selection of the stiffness and damping of a passively suspended vehicle in order to get the best compromise between discomfort, road holding and working space. The final design solution, found by means of the k-ε-optimality approach seems consistent with the solution selected by skilled suspensions specialists. Finally the k-optimality method has proved to be very effective also when applied to complex engineering problems. The optimization of the tyre/suspension system of a sports car has been formulated as a design problem with 18 objective functions. A large set of Pareto-optimal solutions have been computed. Again, the k-optimality approach has proved to be a useful tool for the selection of a fully satisfactory final design solution.

scholarly journals Determining a Robust, Pareto Optimal Geometry for a Welded Joint

2014 ◽  
Vol 1016 ◽  
pp. 39-43
Author(s):  
Simon Barrans ◽  
H.E. Radhi
Keyword(s):  
Welded Joint ◽  
Optimization Problems ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Optimal Geometry ◽  
Pareto Optimal Set ◽  
Particular Solution ◽  
Optimal Set ◽  
Selection Of

Multi-criteria optimization problems are known to give rise to a set of Pareto optimal solutions where one solution cannot be regarded as being superior to another. It is often stated that the selection of a particular solution from this set should be based on additional criteria. In this paper a methodology has been proposed that allows a robust design to be selected from the Pareto optimal set. This methodology has been used to determine a robust geometry for a welded joint. It has been shown that the robust geometry is dependent on the variability of the geometric parameters.

Pareto Optimal Solution Selection for a Multi-Site Supply Chain Planning Problem Using the VIKOR and TOPSIS Methods

2017 ◽  
Vol 8 (3) ◽  
pp. 21-39 ◽  
Author(s):  
Houssem Felfel ◽  
Omar Ayadi ◽  
Faouzi Masmoudi
Keyword(s):  
Supply Chain ◽  
Optimal Solution ◽  
Quality Level ◽  
Planning Problem ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Compromise Solution ◽  
Optimal Solutions ◽  
Pareto Solution ◽  
Solution Approach

In this paper, a multi-objective, multi-product, multi-period production and transportation planning problem in the context of a multi-site supply chain is proposed. The developed model attempts simultaneously to maximize the profit and to maximize the product quality level. The objective of this paper is to provide the decision maker with a front of Pareto optimal solutions and to help him to select the best Pareto solution. To do so, the epsilon-constraint method is adopted to generate the set of Pareto optimal solutions. Then, the technique for order preference by similarity to ideal solution (TOSIS) is used to choose the best compromise solution. The multi-criteria optimization and compromise solution (VIKOR), a commonly used method in multiple criteria analysis, is applied in order to evaluate the selected solutions using TOPSIS method. This paper offers a numerical example to illustrate the solution approach and to compare the obtained results using TOSIS and VIKOR methods.

Solving multiobjective optimal control problems using an improved scalarization method

2020 ◽  
Vol 37 (4) ◽  
pp. 1524-1547
Author(s):  
Gholam Hosein Askarirobati ◽  
Akbar Hashemi Borzabadi ◽  
Aghileh Heydari
Keyword(s):  
Optimal Control ◽  
Uniform Distribution ◽  
Optimal Control Problems ◽  
Pareto Frontier ◽  
Optimal Solution ◽  
Control Problems ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Multiobjective Optimal Control

Abstract Detecting the Pareto optimal points on the Pareto frontier is one of the most important topics in multiobjective optimal control problems (MOCPs). This paper presents a scalarization technique to construct an approximate Pareto frontier of MOCPs, using an improved normal boundary intersection (NBI) scalarization strategy. For this purpose, MOCP is first discretized and then using a grid of weights, a sequence of single objective optimal control problems is solved to achieve a uniform distribution of Pareto optimal solutions on the Pareto frontier. The aim is to achieve a more even distribution of Pareto optimal solutions on the Pareto frontier and improve the efficiency of the algorithm. It is shown that in contrast to the NBI method, where Pareto optimality of solutions is not guaranteed, the obtained optimal solution of the scalarized problem is a Pareto optimal solution of the MOCP. Finally, the ability of the proposed method is evaluated and compared with other approaches using several practical MOCPs. The numerical results indicate that the proposed method is more efficient and provides more uniform distribution of solutions on the Pareto frontier than the other methods, such a weighted sum, normalized normal constraint and NBI.

scholarly journals Multiobjective Optimal Control of HIV Dynamics

2010 ◽  
Vol 2010 ◽  
pp. 1-29 ◽  
Author(s):  
Hassan Zarei ◽  
Ali Vahidian Kamyad ◽  
Sohrab Effati
Keyword(s):  
Optimal Control ◽  
Optimal Solution ◽  
Transcriptase Inhibitor ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Human Immune System ◽  
Weighted Sum Method ◽  
Treatment Interruptions ◽  
Multiobjective Optimal Control

Various aspects of the interaction of HIV with the human immune system can be modeled by a system of ordinary differential equations. This model is utilized, and a multiobjective optimal control problem (MOOCP) is proposed to maximize the CD4+ T cells population and minimize both the viral load and drug costs. The weighted sum method is used, and continuous Pareto optimal solutions are derived by solving the corresponding optimality system. Moreover, a model predictive control (MPC) strategy is applied, with the final goal of implementing Pareto optimal structured treatment interruptions (STI) protocol. In particular, by using a fuzzy approach, the MOOCP is converted to a single-objective optimization problem to derive a Pareto optimal solution which among other Pareto optimal solutions has the best satisfaction performance. Then, by using an embedding method, the problem is transferred into a modified problem in an appropriate space in which the existence of solution is guaranteed by compactness of the space. The metamorphosed problem is approximated by a linear programming (LP) model, and a piecewise constant solution which shows the desired combinations of reverse transcriptase inhibitor (RTI) and protease inhibitor (PI) drug efficacies is achieved.

scholarly journals MODIFICATION OF FIREWORKS METHOD FOR MULTIOBJECTIVE OPTIMIZATION BASED ON NON-DOMINATED SORTING

2019 ◽  
Vol 22 (3) ◽  
pp. 67-78
Author(s):  
A. V. Panteleev ◽  
A. U. Krychkov
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Problems ◽  
Complex Structure ◽  
Optimal Solution ◽  
Future Research ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Equal Importance ◽  
Approximate Result

The article suggests a modification for numerical fireworks method of the single-objective optimization for solving the problem of multiobjective optimization. The method is metaheuristic. It does not guarantee finding the exact solution, but can give a good approximate result. Multiobjective optimization problem is considered with numerical criteria of equal importance. A possible solution to the problem is a vector of real numbers. Each component of the vector of a possible solution belongs to a certain segment. The optimal solution of the problem is considered a Pareto optimal solution. Because the set of Pareto optimal solutions can be infinite; we consider a method for finding an approximation consisting of a finite number of Pareto optimal solutions. The modification is based on the procedure of non-dominated sorting. It is the main procedure for solutions search. Non-dominated sorting is the ranking of decisions based on the values of the numerical vector obtained using the criteria. Solutions are divided into disjoint subsets. The first subset is the Pareto optimal solutions, the second subset is the Pareto optimal solutions if the first subset is not taken into account, and the last subset is the Pareto optimal solutions if the rest subsets are not taken into account. After such a partition, the decision is made to create new solutions. The method was tested on well-known bi-objective optimization problems: ZDT2, LZ01. Structure of the location of Pareto optimal solutions differs for the problems. LZ01 have complex structure of Pareto optimal solutions. In conclusion, the question of future research and the issue of modifying the method for problems with general constraints are discussed.

Combining Convergence and Diversity in Evolutionary Multiobjective Optimization

2002 ◽  
Vol 10 (3) ◽  
pp. 263-282 ◽  
Author(s):  
Marco Laumanns ◽  
Lothar Thiele ◽  
Kalyanmoy Deb ◽  
Eckart Zitzler
Keyword(s):  
Evolutionary Algorithms ◽  
Multiobjective Optimization ◽  
Optimization Problems ◽  
Fundamental Problem ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Pareto Optimal Set ◽  
Multiobjective Optimization Problems ◽  
Baseline Algorithm

Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multiobjective optimization problems, where the goal is to find a number of Pareto-optimal solutions in a single simulation run. Many studies have depicted different ways evolutionary algorithms can progress towards the Pareto-optimal set with a widely spread distribution of solutions. However, none of the multiobjective evolutionary algorithms (MOEAs) has a proof of convergence to the true Pareto-optimal solutions with a wide diversity among the solutions. In this paper, we discuss why a number of earlier MOEAs do not have such properties. Based on the concept of ɛ-dominance, new archiving strategies are proposed that overcome this fundamental problem and provably lead to MOEAs that have both the desired convergence and distribution properties. A number of modifications to the baseline algorithm are also suggested. The concept of ɛ-dominance introduced in this paper is practical and should make the proposed algorithms useful to researchers and practitioners alike.

Analysis of NSGA-II and NSGA-II with CDAS, and Proposal of an Enhanced CDAS Mechanism

2009 ◽  
Vol 13 (4) ◽  
pp. 470-480 ◽  
Author(s):  
Kyoko Tsuchida ◽  
◽  
Hiroyuki Sato ◽  
Hernan Aguirre ◽  
Kiyoshi Tanaka ◽  
...  
Keyword(s):  
Pareto Optimal ◽  
Search Performance ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Nsga Ii ◽  
Degree Of Dominance ◽  
Before And After ◽  
The Relationship ◽  
Best Parameters ◽  
Selection Of

In this work, we analyze the functionality transition in the evolution process of NSGA-II and an enhanced NSGA-II with the method of controlling dominance area of solutions (CDAS) from the viewpoint of front distribution. We examine the relationship between the population of the first front consisting of non-dominated solutions and the values of two metrics, NORM and ANGLE, which measure convergence and diversity of Pareto-optimal solutions (POS), respectively. We also suggest potentials to further improve the search performance of the enhanced NSGA-II with CDAS by emphasizing the parameterS, which controls the degree of dominance by contracting or expanding the dominance area of solutions, before and after the boundary generation of functionality transition. Furthermore, we analyze the behavior of the evolution of the enhanced NSGA-II with CDAS using the best parameters combination and compare its performance with two other algorithms that enhance selection of NSGA-II.

An Orthogonal Multi-objective Evolutionary Algorithm for Multi-objective Optimization Problems with Constraints

2004 ◽  
Vol 12 (1) ◽  
pp. 77-98 ◽  
Author(s):  
Sanyou Y. Zeng ◽  
Lishan S. Kang ◽  
Lixin X. Ding
Keyword(s):  
Evolutionary Algorithm ◽  
Optimization Problems ◽  
Engineering Problem ◽  
Test Problems ◽  
Pareto Optimal ◽  
Large Set ◽  
Multi Objective Optimization ◽  
Multi Objective ◽  
Pareto Optimal Set ◽  
Optimal Set

In this paper, an orthogonal multi-objective evolutionary algorithm (OMOEA) is proposed for multi-objective optimization problems (MOPs) with constraints. Firstly, these constraints are taken into account when determining Pareto dominance. As a result, a strict partial-ordered relation is obtained, and feasibility is not considered later in the selection process. Then, the orthogonal design and the statistical optimal method are generalized to MOPs, and a new type of multi-objective evolutionary algorithm (MOEA) is constructed. In this framework, an original niche evolves first, and splits into a group of sub-niches. Then every sub-niche repeats the above process. Due to the uniformity of the search, the optimality of the statistics, and the exponential increase of the splitting frequency of the niches, OMOEA uses a deterministic search without blindness or stochasticity. It can soon yield a large set of solutions which converges to the Pareto-optimal set with high precision and uniform distribution. We take six test problems designed by Deb, Zitzler et al., and an engineering problem (W) with constraints provided by Ray et al. to test the new technique. The numerical experiments show that our algorithm is superior to other MOGAS and MOEAs, such as FFGA, NSGAII, SPEA2, and so on, in terms of the precision, quantity and distribution of solutions. Notably, for the engineering problem W, it finds the Pareto-optimal set, which was previously unknown.

ECONOMIC ENVIRONMENTAL DISPATCH BY A MODE TECHNIQUE

2008 ◽  
Vol 17 (03) ◽  
pp. 499-512 ◽  
Author(s):  
C. K. PANIGRAHI ◽  
R. CHAKRABARTI ◽  
P. K. CHATTOPADHYAY
Keyword(s):  
Optimization Problem ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Multi Objective Optimization ◽  
Multi Objective ◽  
Pareto Optimal Set ◽  
Sample Test ◽  
Optimal Set ◽  
Mode Technique

Multi-objective differential evolution (MODE) is proposed to handle economic environmental dispatch problem, which is a multi-objective optimization problem (MOOPs) with competing and noncommensurable objectives. The proposed approach has a good performance in finding a diverse set of solutions and in converging near the true Pareto-optimal set. Numerical results for two sample test systems have been presented to demonstrate the capabilities of the proposed approach to generate well-distributed Pareto-optimal solutions of economic environmental dispatch problem in one single run. Simulation results with the proposed approach have been validated with Non-dominated Sorting Genetic Algorithm-II.

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